Paul Erdős’s Five Hundred Dollars

Around 1995, an eighty-two-year-old man decided to raise the stakes. The problem he had posed in 1946 had resisted for half a century. He had promised three hundred dollars to whoever solved it. He raised the prize to five hundred. It was his way of saying something simple: I believe so strongly that I am right that I will pay you to prove me wrong.

Paul Erdős died the following year. Nobody had claimed the money. He died convinced he had been right.

He was wrong. We have known that since last month. And the one who demonstrated it was not a mathematician.

The man who paid for his questions

Erdős is a singular figure in the history of science. Hungarian, born in 1913, he spent his life without a home, without a permanent position, owning almost nothing, traveling from university to university with a suitcase, sleeping at other people’s houses, living for mathematics alone. He posed nearly fifteen hundred problems. He attached a sum of money to many of them, and the amount signaled the difficulty. A few dollars for an approachable problem, several hundred for those that taunted him.

He really did pay. One day, while driving to a town in Georgia, a colleague mentioned a theorem he had just proved. Erdős asked whether he had offered a prize for it. The two men made a detour to the library to check. The prize existed: one hundred dollars. Erdős paid. A rather expensive taxi ride, someone pointed out. He found that very funny.

The five-hundred-dollar problem I am telling you about has a forbidding name: the unit distance problem. Yet you can explain it on the corner of a table.

The problem on a sheet of paper

Put some points on a sheet of paper. Count the pairs of points separated by exactly the same distance, say one centimeter.

Two points, one pair. An equilateral triangle, three pairs. You add points and look for the best arrangement, the one that produces the greatest number of pairs exactly one centimeter apart.

Erdős’s question fits into one sentence. When there are millions of points, which arrangement beats all the others?

For eighty years, the answer seemed settled. A grid. Graph paper, with points at the intersections. Regular, dense, efficient. Erdős had proved that a grid could achieve a certain number of pairs, and he bet nobody would do significantly better. The whole field followed him. The best specialists ran up against it for decades, each arriving at the same bounds by different routes, without ever glimpsing an opening.

Last month, an OpenAI model showed that it is possible to do better than the grid. Not by chance, in a particular case. Better in general, with a construction that holds as the number of points grows. Gowers, a Fields medalist, said that if a human had written this proof and submitted it to a journal, he would have voted to publish it without hesitation.

Why nobody had found it

This is where the story becomes interesting, where it stops being about mathematics and starts being about us.

Imagine everyone is looking for treasure in a field. For eighty years, people have dug in the field, because the treasure is in the field, obviously. The model went to dig in the neighboring forest. The treasure was in the forest.

The forest has a name: algebraic number theory. A region of mathematics that seems to have nothing to do with points on a sheet of paper. Nobody had looked there because nothing suggested they should.

Thomas Bloom, the mathematician who maintains the list of Erdős problems, explained what a single person would have needed to bring together to find the answer. Four things that never coexist. Devoting an unreasonable amount of time to it. Trying to prove Erdős wrong when the entire profession believed he was right, meaning months of swimming against the current. Having the outlandish idea of looking in the forest. And knowing that distant forest well enough to find your way around it.

A person who meets all four conditions does not exist. Or if they do, they have better things to do with their career than bet against half a century of consensus. You do not sacrifice three years of your life digging in the forest when everyone swears the treasure is in the field. It is a bad calculation, and humans are animals that calculate.

The machine does not calculate that way. It has no career, no ego that bleeds, no reputation to uphold among peers. It dug in the forest because it digs everywhere, with no preconception about where the treasure ought to lie. It did not know you were not supposed to look there.

What this discovery damages

We are deeply attached to a certain story about discovery. The genius has a flash of insight. The apple falls, Newton understands. The superior mind suddenly perceives what others have failed to see. We love this story because it makes the great mind an exception of nature, and intuition a spark that can neither be explained nor reproduced.

This machine puts a dent in that story.

It had no revelation. It did not glimpse the truth in a flash. It did the opposite. It tirelessly explored paths we had ruled out in advance. It pushed the door everyone thought was closed, simply because it was a door and it tries them all.

Hence the question that has haunted me since I read about this. What if intuition had never been magical? What if the great mathematician’s instinct was also just exploration, taken further and faster than the next person’s, but made of the same stuff? Perhaps we invented the idea of a gift to avoid facing a more modest truth. Genius does not see what others cannot see. It digs a little longer before giving up.

Look at what happened here. The problem did not resist for eighty years because it was too difficult for the human mind. It resisted because no brain could, or would, explore such an unpromising direction for so long. It was not a matter of talent. It was a matter of patience and lifespan. We die too soon. We grow weary. We need our bets to pay off before the end. We dread the humiliation of being wrong in front of others. The machine knows none of these constraints. It does not tire, does not die halfway through a calculation, feels no shame.

What we took for the nobility of our minds may have been only the shadow of our limitations.

What I have left

I do not conclude that the human mind is worthless. I conclude that one of its glories, sovereign intuition, was probably a legend we told about patient work disguised as a flash of insight.

I have one consolation left, and I do not know how long it will hold. The machine searched the forest, but a man showed it the field. It solved the problem, but it was Erdős, half a century earlier, who judged that problem worthy of a lifetime and five hundred dollars. Choosing the question, deciding it is worth wearing yourself out over, has not yet fallen into the hands of machines. Whether that choice really comes down to taste and meaning, or is itself just another exploration we have not finished mechanizing, remains to be seen.

I do not have the answer. I only think the question deserves more than a shrug. Because every time a machine clears a step, we move the step. Yesterday it failed at middle-school exercises, today it refutes Erdős, and already some people are explaining that it does not really count, that this is not thinking. We have moved the bar so often we no longer know where it is. It is an elegant way of never looking ourselves in the face.

An elegant proof remains elegant whether it comes from a brain or a graphics card. Truth does not care who found it. We do. And the unease we feel on learning that the machine dug where our genius had not even thought of taking a shovel, I believe it is infinitely better to face it than to dismiss it by repeating that it was not so difficult.

Erdős would have paid the five hundred dollars. He would have thought it a wonderful story, and found the detour to the library even funnier than the previous one.

See You Next Week…

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